How to scale two objects to fit together
Assuming nothing is centered or aligned (but rotation is consistent), use Snapping >> Vertex >> Active.

1.) Select a bottom vertex with which to align the half-sphere. L to select connected vertices.

2.) Drag it over to snap it to its "sister" vertex.

3.) Reselect the vertex and Shift + S to move Cursor to Selected.  Change the Pivot Point to 3D Cursor.

4.) Now, select the opposite vertex (make active) and L to select connected vertices.

5.) Scale to "sister" vertex.